Coquaternions, Metric Invariants of Biologic Systems and Malignant Transformations

Different hypotheses of carcinogenesis have been proposed based on local factors and mechanisms. It is assumed that changes of the metric invariants of a biologic system (BS) determine common functional mechanisms of cancer growth. Numerous data demonstrate an existence of three invariant feedback patterns of BS: negative feedback (NFB), positive feedback (PFB) and reciprocal links (RL). These patterns algebraically represent basis elements of a Lie algebra sl(2,R) and imaginary part of coquaternion. Considering coquaternion as a model of a functional core of a BS, conditions of the system can be identified with the points of three families of hypersurfaces in R42: hyperboloids of one sheet, hyperboloids of two sheets and double-cones. Corresponding quadratic form relates entropy contributions of basis elements to the energy level of the system, so that anabolic states of the system will correspond to the points of a hyperboloid of one sheet, while catabolic conditions to the points of a hyperboloid of two sheets. Equilibrium states will lie in a double cone. Hypothetically anabolic and catabolic states dominate intermittently oscillating around the equilibrium. Deterioration of basis elements will cause domination of catabolic processes and cancer development demonstrating the tendency of the malfunctioning system to remain inside the double cone.